Large-system factorial-growth conjecture for explosive birth processes

Let $F_1=\ldots=F_A$ be the common birth-rate function for $A$ agents, and let the initial state be

$$X(0)=(1,\ldots,1).$$

For each $a\geq 2$, let $c(A,a)$ denote the tail-dependence constant appearing in the symmetric asymptotic relation for $a$ agents. Large-system factorial-growth conjecture. For every $a\geq 2$,

$$\lim_{A\to\infty}c(A,a)=a!,$$

so that

$$\lim_{A\to\infty}\frac{c(A,a)}{c(A,a-1)}=a.$$

The conjecture predicts factorial, and hence super-exponential, growth of the dependence constants in the number of agents conditioned to be losers. It is motivated by numerical computations for $F(k)=k^2$ and by the heuristic following the conjecture; the supplied source does not state a proof or a resolution.